Unitarity and irreducibility of the limits of discrete series

Put $u_t=\left(\begin{array}{cc}1& t\\ 0& 1\end{array}\right)$ and ${\mathrm{\ell }}_t=\left(\begin{array}{cc}1& 0\\ t& 1\end{array}\right)$; their real infinitesimal generators are $N_{+}$ and $N_{-}$ from [F5]. If $(b_1,b_2)$ is the bottom row of $k_{\theta }g$, the Iwasawa cocycle in [F1] gives $(\mathrm{\Pi }(g)f_m)(k_{\theta })=(b_2-i{b}_1{)}^m(b_1^2+b_2^2{)}^{-(m+1)\mathrm{/}2}$. For odd $m$, both exponents are integers. When $g=u_t$ or $g={\mathrm{\ell }}_t$, $b_1,b_2$ are affine in $t$; at $t=0$, $b_1^2+b_2^2=1$ and $b_2-i{b}_1=e^{i\theta }$. Compactness of $K$ gives a complex neighborhood of $t=0$, uniform in $\theta$, on which these factors are analytic and their denominators stay nonzero. Thus both orbit maps have power series converging uniformly on $K$, hence in $H$, and their Taylor coefficients are $L_{N_{+}}^k{f}_m\mathrm{/}k!$ or $L_{N_{-}}^k{f}_m\mathrm{/}k!$. By [F3] and [F5], these coefficients remain in the same algebraic positive or negative tail as $f_m$, so each small-$t$ orbit vector lies in the corresponding closed span $D_1^{+}$ or $D_1^{-}$. Unitarity in [F1] extends this inclusion from finite sums to each closure, and the inverse elements give equality. Every $u_t$ and ${\mathrm{\ell }}_t$ is a product of elements with sufficiently small parameter. Moreover, for $s\mathrm{\ne }0$, $u_s{\mathrm{\ell }}_{-1\mathrm{/}s}{u}_s=\left(\begin{array}{cc}0& s\\ -s^{-1}& 0\end{array}\right)$ and $\left(\begin{array}{cc}0& s\\ -s^{-1}& 0\end{array}\right){\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)}^{-1}=\mathrm{diag}(s,s^{-1})$. Hence the two unipotent subgroups generate every determinant-one matrix: if $g=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ has $a\mathrm{\ne }0$, then $g={\mathrm{\ell }}_{c\mathrm{/}a}\mathrm{diag}(a,a^{-1})u_{b\mathrm{/}a}$; if $a=0$, then $c\mathrm{\ne }0$ and $u_{1}g$ has nonzero upper-left entry. Thus both closed spans are G-invariant.

Let $W$ be a nonzero closed G-invariant subspace of $D_1^{+}$. Since step 1.1 proves $D_1^{+}$ is G-invariant, $W$ is K-invariant. Choose $0\mathrm{\ne }v\in W$. By [F2], $v$ has an orthogonal expansion in the lines $\mathbb{C}{f}_{1+2j}$, so some K-character projection $P_{j}v$ is nonzero. Its degree-one character integral from [F4] is a norm limit of sums of K-translates of $v$, all in $W$; closedness gives $f_{1+2j}\in W$ for some $j$. For real $X\in \mathfrak{g}$, the difference quotients $(\mathrm{\Pi }(\exp (tX))w-w)\mathrm{/}t$ for any smooth $w\in W$ lie in $W$ and converge in norm to $L_{X}w$; complex linearity gives stability under $E_{\pm }$. The K-type vectors are smooth. By [F3], $L_{E_{+}}$ raises every positive-tail weight with coefficient $j+1\mathrm{\ne }0$, while $L_{E_{-}}$ lowers it with coefficient $-j\mathrm{\ne }0$ for $j>0$; the boundary coefficient at $j=0$ is zero. Iteration therefore gives every $f_{1+2k}\in W$. Their span is dense by [F2], so $W=D_1^{+}$.

Let $W$ be a nonzero closed G-invariant subspace of $D_1^{-}$. Choose $0\mathrm{\ne }v\in W$. By [F2], its orthogonal expansion in the lines $\mathbb{C}{f}_{-1-2j}$ has a nonzero coefficient; the corresponding K-character projection [F4] is a norm limit of K-translates in $W$, so $f_{-1-2j}\in W$ for some $j$. The difference-quotient argument of step 2.1 gives stability under $E_{\pm }$. If $j>0$, the nonzero coefficient $-j$ in $L_{E_{+}}{f}_{-1-2j}$ moves up to the boundary weight $-1$; from there the nonzero coefficients $j+1$ in $L_{E_{-}}$ generate every lower weight. Thus every $f_{-1-2k}$ lies in $W$, and density [F2] gives $W=D_1^{-}$. Hence both limits are irreducible.

By [F1] and step 1.1, the restrictions to the closed limits are strongly continuous unitary representations. Their orthogonal direct sum is $I_{1,0}$ by [F2]; their K-type multiplicities and weights are [F2], and their Casimir scalar is [F3].